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Improved Bounds for the Unsplittable Flow Problem
 In Proceedings of the 13th ACMSIAM Symposium on Discrete Algorithms
, 2002
"... In this paper we consider the unsplittable ow problem (UFP): given a directed or undirected network G = (V, E) with edge capacities and a set of terminal pairs (or requests) with associated demands, find a subset of the pairs of maximum total demand for which a single flow path can be chosen for eac ..."
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Cited by 56 (6 self)
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In this paper we consider the unsplittable ow problem (UFP): given a directed or undirected network G = (V, E) with edge capacities and a set of terminal pairs (or requests) with associated demands, find a subset of the pairs of maximum total demand for which a single flow path can be chosen for each pair so that for every edge, the sum of the demands of the paths crossing the edge does not exceed its capacity.
Graph decomposition and a greedy algorithm for edgedisjoint paths
 In Proceedings of the 15th ACMSIAM Symposium on Discrete Algorithms (SODA
, 2004
"... Abstract Given a directed graph G = (V, E) with n vertices and a parameter l> = 1, we present an algorithm that finds a cut (set of edges) of size O((n2/l2)log2(n/l)) whose removal separates every pair of vertices (s,t) in G such that the minimum distance between s and t in G is at least l. This ..."
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Cited by 28 (0 self)
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Abstract Given a directed graph G = (V, E) with n vertices and a parameter l> = 1, we present an algorithm that finds a cut (set of edges) of size O((n2/l2)log2(n/l)) whose removal separates every pair of vertices (s,t) in G such that the minimum distance between s and t in G is at least l. This theorem implies a nearly tight analysis of the greedy algorithm for finding edgedisjoint paths in directed graphs, and gives the best known approximation factor for this problem in terms of the number of vertices.
Directed Metrics and Directed Graph Partitioning Problems
 In Proc. of SODA, 2006
"... The theory of embeddings of finite metrics has provided a powerful toolkit for graph partitioning problems in undirected graphs. The connection comes from the fact that the integrality gaps of mathematical programming relaxations for sparsest cut in undirected graphs is exactly equal to the minimum ..."
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Cited by 10 (1 self)
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The theory of embeddings of finite metrics has provided a powerful toolkit for graph partitioning problems in undirected graphs. The connection comes from the fact that the integrality gaps of mathematical programming relaxations for sparsest cut in undirected graphs is exactly equal to the minimum distortion required to embed certain metrics into ℓ1. No analog of this metric embedding theory exists for directed (asymmetric) metrics, the natural distance functions that arise in considering mathematical relaxations for directed graph partioning problems. We initiate a study of metric embeddings for directed metrics, motivated by understanding directed variants of sparsest cut. It turns out that there are two different ways to formulate sparsest cut in directed graphs (depending on whether one insists on partitioning the graph into two pieces or not). Different subclasses of directed metrics arise in the consideration of mathematical relaxations for these two formulations and the embedding questions that result are quite different. Unlike in the undirected case, where the natural host space is ℓ1, the host space in the directed case is not obvious and depends on the problem formulation. Our work is a first step at understanding this space of directed metrics, the resulting embedding questions and their relationships to directed graph partitioning problems. 1
Combinatorial algorithms for the unsplittable flow problem
 Algorithmica
"... We provide combinatorial algorithms for the unsplittable flow problem (UFP) that either match or improve the previously best results. In the UFP we are given a (possibly directed) capacitated graph with n vertices and m edges, and a set of terminal pairs each with its own demand and profit. The obje ..."
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Cited by 10 (3 self)
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We provide combinatorial algorithms for the unsplittable flow problem (UFP) that either match or improve the previously best results. In the UFP we are given a (possibly directed) capacitated graph with n vertices and m edges, and a set of terminal pairs each with its own demand and profit. The objective is to connect a subset of the terminal pairs each by a single flow path subject to the capacity constraints such that the total profit of the connected pairs is maximized. We consider three variants of the problem. First is the classical UFP in which the maximum demand is at most than the minimum edge capacity. It was previously known to have an O ( √ m) approximation algorithm; the algorithm is based on the randomized rounding technique and its analysis makes use of the Chernoff bound and the FKG inequality. We provide a combinatorial algorithm that achieves the same approximation ratio and whose analysis is considerably simpler. Second is the extended UFP in which some demands might be higher than edge capacities. Our algorithm for this case improves the best known approximation ratio. We also give a lower bound that shows that the extended UFP is provably harder than the classical UFP. Finally, we consider the bounded UFP in which the maximum demand is at most 1 K times the minimum edge capacity for some K> 1. Here we provide combinatorial algorithms that match the currently best known algorithms. All of ours algorithms are strongly polynomial and some can even be used in the online setting. 1
Approximation Algorithms and Hardness Results for Cycle Packing Problems
"... The cycle packing number νe(G) of a graph G is the maximum number of pairwise edgedisjoint cycles in G. Computing νe(G) is an NPhard problem. We present approximation algorithms for computing νe(G) in both undirected and directed graphs. In the undirected case we analyze ..."
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Cited by 9 (1 self)
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The cycle packing number νe(G) of a graph G is the maximum number of pairwise edgedisjoint cycles in G. Computing νe(G) is an NPhard problem. We present approximation algorithms for computing νe(G) in both undirected and directed graphs. In the undirected case we analyze
An O( √ n)Approximation Algorithm For Directed Sparsest Cut
"... We give an O (√ n)approximation algorithm for the Sparsest Cut Problem on directed graphs. A naïve reduction from Sparsest Cut to Minimum Multicut would only give an approximation ratio of O (√ n log D), where D is the sum of the demands. We obtain the improvement using a novel LProunding method f ..."
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Cited by 7 (1 self)
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We give an O (√ n)approximation algorithm for the Sparsest Cut Problem on directed graphs. A naïve reduction from Sparsest Cut to Minimum Multicut would only give an approximation ratio of O (√ n log D), where D is the sum of the demands. We obtain the improvement using a novel LProunding method for fractional Sparsest Cut, the dual of
On Consistent Migration of Flows in SDNs
, 2016
"... Abstract—We study consistent migration of flows, with special focus on software defined networks. Given a current and a desired network flow configuration, we give the first polynomialtime algorithm to decide if a congestionfree migration is possible. However, if all flows must be integer or are u ..."
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Abstract—We study consistent migration of flows, with special focus on software defined networks. Given a current and a desired network flow configuration, we give the first polynomialtime algorithm to decide if a congestionfree migration is possible. However, if all flows must be integer or are unsplittable, this is NPhard to decide. A similar problem is providing increased bandwidth to an application, while keeping all other flows in the network, but possibly migrating them consistently to other paths. We show that the maximum increase can be approximated arbitrarily well in polynomial time. Current methods as RSVPTE consider unsplittable flows and remove flows of lesser importance in order to increase bandwidth for an application: We prove that deciding what flows need to be removed is an NPhard optimization problem with no PTAS possible unless P = NP. I.